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13    MEASURES WITH BASE //        181

this linear form is a (complex) measure on X, because if K is any compact
subset of X, we have

i r

fe

I \\f\\

for all/e Jfc(X; K), by virtue of (13.10.3). The measure so defined is called
the measure with density g relative to /*, and is denoted by g  ju; when g is
continuous, this agrees with the definition (13.1.5). Measures of the form
g - n are also called measures with base //. It follows immediately from this
definition that if g takes values in R (resp. ^0 almost everywhere), then the
measure g  u is real (resp. positive). Furthermore, g  fi does not change if we
replace g by an equivalent function (with respect to u), and therefore we may
restrict ourselves to the case where g is everywhere finite and universally
measurable (13.9.12).

If g and g2 are two locally integrable functions, then g^ + g2 and agt
are locally integrable (a being any complex scalar), and we have

(13.13.2)      fa + ff2)  \i = #! -11 + g2 ' n,       (agj - fi

For every complex-valued locally integrable function g, we have

(13.13.3)    #  /* =   //,       (g-fi = (&g) ' I*,       Sfa - p) * (Sg)

(13.13.4) The set &\^ R(X, u) (resp. &\^ C(X, JM)) of real-valued (resp.
complex-valued) locally ^-integrable functions is a real (resp. complex)
vector space, often denoted by J?loc(X, /*) or J?IOO(A) or JS? loc(X) or ^,oc ,
For every compact subset K of X, the mapping pK : g\-+ f |^K| rf/i is a
semi-norm on this space. We shall always suppose jSf ,oc to be endowed with the
topology defined by these seminorms. If (Kw) is an increasing sequence of
compact subsets of X, such that (J Kn = X and Kn c Kn + 1 (3.18.3), then it is

n

immediately seen that the topology of & ,oc is defined by the seminorms
pKn. In particular, if X is compact, then jS?/OCt R(X, fi) (resp. j?,OCp C(X, ju)) is
identical with JS?i(X, //) (resp. ^c(x j"))- The 'set of locally integrable real-
valued functions g such that pK(g) = 0 for all compact subsets K of X is the
space Jf of ^-negligible functions. We define

and L/OCj C(X, p) analogously. The seminorms p^(g) depend only on the
class g of g: if we put pK(g) = p^g), then we obtain seminorms defining the
topologies of these spaces, which are therefore metrizable and locally convex. a locally ^-integrable function. Since /H* \fg d^ is defined on the